Remark 2

Let $(X_j)_{j\in J}$ be a family of topological spaces. Let X denote the product topological space $\prod_{j\in J}X_j$ and let $x= (x_j)$ be a point of X.

For every $i\in J$, let $u_i: X_i\rightarrow X$ be the mapping such that, for $z\in X_i$, pr$_i\circ u_i(z) =z$ and, for every $j\in J$ distinct from $i$, pr$_j\circ u_i(z) =$ $x_j$. The mapping $u_i$ is continuous and the mapping $\pi_1(u_i, x_i)$ is identified with the canonical injection of the factor $\pi_1(X_i, x_i)$ into the product group of the family $(\pi_1(X_j, x_j))_{j\in J}($cf. A, I, p. 45).

Suppose that the set J is finite and, for every $j\in J$, let $\gamma_j$ be an element of $\pi_1(X_j, x_j)$. The element $(\gamma_j)$ of $\pi_1(X, x)$ is the composite of the classes of loops $(u_j)_*(\gamma_j),j\in J$, these classes being pairwise permutable.